Event A occurs with probability 0.1 and event B with probability 0.5

A) What is the maximum probability that the intersection of A and B can have?
B) What is the minimum probability that the intersection of A and B can have?
C) If it is know that P(A complement and B complement) = .47, find P(A or B) and P(A and B)
D) Draw a Venn diagram based on part c
E) Using parts can d d, find Probability of A given B.

Thanks!

A) The maximum probability that the intersection of A and B can have is the smaller of the two individual probabilities. In this case, it would be 0.1.

To get this answer, you need to know the basic rule of probabilities that states that the probability of the intersection of two events (A and B) cannot be greater than the probability of each individual event. Since event A occurs with probability 0.1 and event B occurs with probability 0.5, the maximum probability for their intersection is 0.1.

B) The minimum probability that the intersection of A and B can have is the sum of the probabilities of A and B minus 1. In this case, it would be 0.6 - 1 = -0.4. However, probabilities cannot be negative, so the minimum probability is 0.

To get this answer, you use the same basic rule of probabilities mentioned above. The sum of the probabilities of A and B is 0.1 + 0.5 = 0.6. Subtracting 1 from that gives you -0.4, but since probabilities cannot be negative, the minimum probability is 0.

C) To find P(A or B), you need to find the probability of event A or event B occurring. This can be calculated using the formula P(A or B) = P(A) + P(B) - P(A and B). However, we don't have the value for P(A and B).

To find P(A and B), we can use the formula P(A and B) = P(A) + P(B) - P(A or B), rearranged to solve for P(A and B):

P(A and B) = P(A) + P(B) - P(A or B)
P(A and B) = 0.1 + 0.5 - P(A or B)

Given that P(A complement and B complement) = 0.47, we can use a similar formula to find P(A or B complement):

P(A or B complement) = P(A) + P(B complement) - P(A and B complement)
0.47 = 0.1 + (1 - 0.5) - P(A and B complement)
0.47 = 0.6 - P(A and B complement)

Combining the two equations, we have:

0.47 = 0.1 + 0.5 - P(A or B) + 0.6 - P(A and B complement)

Simplifying and rearranging, we get:

P(A and B complement) - P(A or B) = 0.73

Since P(A and B complement) + P(A and B) = P(A) = 0.1, we can solve for P(A and B complement) and substitute it into the equation above to find P(A or B):

0.1 - P(A and B) - P(A or B) = 0.73
0.1 - P(A and B) - P(A or B) + P(A or B) = 0.73 + P(A or B)
0.1 - P(A and B) = 0.73 + P(A or B)
0.1 - 0.73 = P(A or B) - P(A and B)
-0.63 = P(A or B) - P(A and B)

This equation tells us that P(A or B) - P(A and B) is equal to -0.63. However, probabilities cannot be negative, so we can conclude that P(A or B) - P(A and B) would be 0, making P(A or B) = P(A and B). So, the probability of A or B is equal to the probability of A and B.

Therefore, P(A or B) = P(A and B) = 0.1.

D) Since P(A or B) = P(A and B) = 0.1, the Venn diagram will have overlapping circles where the intersection region is equal to the individual probability of 0.1.

E) To find the probability of A given B (P(A|B)), we can use the formula P(A|B) = P(A and B) / P(B).

Since P(A and B) = 0.1 and P(B) = 0.5, we can substitute these values into the formula:

P(A|B) = 0.1 / 0.5
P(A|B) = 0.2

Therefore, the probability of A given B is 0.2.